Question 2.2.9: Estimating Velocity Numerically Suppose that a sprinter reac...
Estimating Velocity Numerically
Suppose that a sprinter reaches the following distances in the given times. Estimate the velocity of the sprinter at the 6-second mark.
| t(sec) | 5.0 | 5.5 | 5.8 | 5.9 | 6.0 | 6.1 | 6.2 | 6.5 | 7.0 |
| f(t) (ft) | 123.7 | 141.01 | 151.41 | 154.9 | 158.4 | 161.92 | 165.42 | 175.85 | 193.1 |
The instantaneous velocity is the limit of the average velocity as the time interval shrinks. We first compute the average velocities over the shortest intervals given, from 5.9 to 6.0 and from 6.0 to 6.1.
Since these are the best individual estimates available from the data, we could just split the difference and estimate a velocity of 35.1 ft/s. However, there is useful information in the rest of the data. Based on the accompanying table, we can conjecture that the sprinter was reaching a peak speed at about the 6-second mark. Thus, we might accept the higher estimate of 35.2 ft/s. We should emphasize that there is not a single correct answer to this question, since the data are incomplete (i.e., we know the distance only at fixed times, rather than over a continuum of times).